I've found a solution to my problem, but am curious why my original plan didn't work. Very curious, as I spent hours on it.

The setup:
My thermostat is not connected to or related to HVAC. I'm using a thermostat to turn on a ventalation fan when my set temperature hits. Without drawing it out, it's like this:
- 24v to t-stat
- 24v from t-stat to relay switch (when t-stat is closed)
- When activated, relay switch closes 110v circuit to power on my fan
- Thermostat is: Non-programmable. Honeywell CT31A (also tried a diff t-stat with same results)

My original plan:
Seemed very simple. Just needed two wires up to my t-stat.
- WireA -> 24v lead to R in t-stat
- WireB -> Connect to Y back down to my relay switch

I wired that up. Moved the t-stat to a lower setting to simulate temperate increase. Just as expected, the fan kicked on. (smile) "Perfect.", I thought. So I slide the t-stat back to a higher temp to shut the fan back off. ...Dang thing kept running. I thought maybe there's a short timer in the t-stat, so I waited. ...and waited.

My question:
Why does the fan not shut off? 24v just keeps going down to my relay switch until I turn the t-stat from 'cool' to 'off'. There seems to be something in the t-stat circuit board that's expecting something else to happen (in addition to the t-stat switch opening).

How I solved it:
I finally got frustrated and bypassed the circuit board. This particular t-stat model made that very easy to do, as it had screw holes on the back of the front panel. So I just connected WireA to R on the front panel and WireB to Y on the front panel. Any "thinking" or logic introduced by the circuit board is bypassed.

I've attached a few pics for reference.
Thanks for your input.

 

IT has an "anticipator" resistor that tries to add a level of intelligence to the switch? Have you tried to adjust it for minimum effect?

 

Thanks for the reply telecom guy.
I didn't try that. As I understand it, the anticipator affects when the temp-controlled switch flips by changing resistance. Meaning it happens at "the front panel" (pic one just right of the coil) and not on the circuit board. Given I can wire directly to the switch and get good results, I ruled that out.
Maybe my logic is wrong though.